Proposition 6 (Tight Decomposition Functions for Mixed-Monotone Systems) [1]},[2]} 
For any system in the form of (REF )  and \(\mathbb {I}\mathcal {Z} \triangleq [\underline{z},\overline{z}]\) , a tight (optimal)  discrete-time or continuous-time mixed-monotone decomposition function \(f^O_d=[f^O_{d,1} \dots f^O_{d,n_x}]\)   and its corresponding tight inclusion function \(T^{f_d}_O(\mathbb {I}\mathcal {Z})\triangleq [f^O_d(\underline{z},\overline{z}),f^O_d(\overline{z},\underline{z})]\)  (i.e., \(T^f_O(\mathbb {I}\mathcal {Z})=T^{f_d}_O(\mathbb {I}\mathcal {Z})\) ) can be  constructed as follows:
